As with the previous lessons, this lesson begins with a review of the prior day’s learning, but in this case, the teacher extends the review by anticipating a possible procedural error related to place value that the students might make when using the base tens blocks to represent bridging with decimals, and explaining how to avoid this error. Rosenshein notes that effective teachers anticipate student errors and explicitly teach students how to avoid them.

The teacher follows the same format of introducing the focus of the day’s lesson and then moving through extended sessions of both guided and independent practice. By today’s lesson, it is possible to notice that the students’ fluency in using the strategy of bridging has increased. For example, they are often able to dispense with concrete representations and move directly to visual or even abstract representations. The students in one small group are also seen to be readily able to extrapolate the mathematical equation from a word problem without help, a process that the teacher has modelled throughout the week.

The teacher concludes this final lesson of the week with a plenary session during which she asks the students how they feel about what they have learned throughout the week. She asks what they enjoyed and what they feel more competent in doing, as well as asking them to compare their confidence about maths at the beginning of the week with their confidence after a week of lessons. This helps contribute towards the students’ developing mathematical identities, as well as encouraging them to be metacognitive about their own learning. It is rewarding to observe how positively and enthusiastically the students express themselves when talking about their learning.

Reviewing prior learning

The evidence-informed strategies you will observe in this video include:

• Using mathematical language to consolidate conceptual understanding
• Using fully worked examples to supports students’ procedural understanding
• Asking questions to check student understanding
• Guiding student practice to build conceptual and procedural understanding
• Integrating multiple representations to support understanding and flexible thinking

This first session of the lesson is dedicated to revisiting the previous day’s learning, with a particular focus on a potential procedural error that the teacher has anticipated. Observing a student’s practice the previous day had reminded her of the importance of emphasising place value knowledge when using base tens blocks to bridge with decimals. She models an example, using place value labels to mark the different columns in the concrete representation in order to ensure that the students understand the need to move the blocks from one column to the next when they bridge through a multiple of one. She then has the class practise this procedure to consolidate this understanding. Note that during this part of the session, they move from a concrete representation of the problem with manipulatives directly to an abstract representation (a balancing equation), which demonstrates the students’ developing fluency.

Learning to bridge when subtracting decimals

In this video you will observe the teacher:

• Guiding student practice to support efficient learning
• Asking questions to check students’ procedural knowledge
• Checking for student understanding to consolidate learning
• Using prior knowledge to minimise cognitive load and strengthen learning
• Working in small steps to support effective learning
• Asking questions to consolidate students’ mathematical knowledge

This part of the lesson begins with the students using base tens blocks to learn to bridge when subtracting decimals. The teacher revisits their prior learning by reminds them of the principle they used earlier in the week when using materials to represent subtraction problems, which is that they don’t build the number to be subtracted with the materials. She also reminds them to start from the right when subtracting on a number line. These reminders help to reduce the students’ cognitive load so that they can focus their limited working memory capacity on the mathematical problems.

In this video, you will see the teacher ask a small group of students if they are confident to work out the practice problems without using materials, and they confirm that they are. Later in the lesson, the teacher goes straight to asking the class to solve a problem using a balancing equation, without using either materials or a number line. This demonstrates that the extended practice with multiple representations throughout the week has built the students’ conceptual and procedural understanding and developed their mathematical fluency.

At the end of the video, the teacher asks two students to explain their thinking and their mathematical working so that she can check their understanding. She also asks how confident they are feeling, and affirms that they are ‘still learning’ when they express less than total confidence. This is important for their developing mathematical identities, because it helps them to see learning maths as a process rather than something they either can or cannot do.

Word problems

The main strategies to note is this video are:

• Focusing on word problems to consolidate mathematical understanding and procedural knowledge
• Building students’ mathematical self-concept by affirming different methods of reaching the correct solution

In this short video, the teacher uses a word problem to allow the students to apply their use of bridging strategies. Thanks to the teacher’s modelling throughout the week, the students are easily able to extrapolate the mathematical equation from the word problem, and solve it using a number line without needing to use materials. The teacher demonstrates two slightly different methods used by different students in the group, explicitly stating that they have both reached the correct answer despite using a different number of jumps on the number line, which affirms both students in their mathematical knowledge.

Building students’ mathematical identities

In this video, you will notice the teacher:

• Developing and affirming the students’ mathematical identities and academic self-concept by encouraging them to think about their own learning

In this final video of the week, the teacher concludes the week’s maths lessons by asking the class to reflect on their learning. The consider what went well, and what they feel confident about. Some students mention the types of problems they feel more confident in solving (such as working with fractions and decimals), while another describes how using materials has helped improve her understanding and confidence. Importantly, the teacher prompts the students to be metacognitive about their learning by asking them to compare how they felt about working with mathematical problems involving fractions and decimals at the beginning of the week with how they feel about them after a week of teaching and extensive practice. This ability to evaluate their own progress and learning is an important part of the students developing a positive mathematical identity. Their enthusiastic and positive responses are a good indication of how enjoyable and motivating this systematic approach to teaching maths for mastery can be for students.